What are the mathematical prerequisites for an AI researcher?
Good Mathematics Foundation
Begin by ensuring full competency with intermediate algebra and some other foundations of calculus and discrete math, including the terminology and basic concepts within these topics.

Infinite series
Logical proofs
Linear algebra and matrices
Analytic geometry, especially the distinction between local and global extremes (minima and maxima), saddle points, and points of inflection
Set theory
Probability
Statistics

Foundations of Cybernetics
Norbert Wiener, Cybernetics, 1948, MIT Press, contains time series and feedback concepts with clarity and command not seen in subsequent works; it also contains an introduction to information theory beginning with Shannon's log2 formula for a defining the amount of information in a bit. This is important to understand the expansion of the information entropy concept.
Calculus
Find a good calculus book and make sure you have clarity around key theory and application in these categories.

Time series
Infinite series
Convergence — Artificial networks ideally converge to an optimum during learning.
Partial differentials
Jacobian and Hessian matrices
Multivariate math
Boundary regions
Discrete math

Much of that is in Calculus, Strang, MIT, Wellesley-Cambridge Press. Although the PDF is available on the web, it is basic and not particularly deep.  The one in our laboratory's library is Intermediate Calculus, Hurley, Holt Rinehart &amp; Winston, 1980. It is comprehensive and in some ways better laid out than the one I have in my home library, which Princeton uses for sophomores.
Ensure you are comfortable working in spaces beyond ℝ2 (beyond 2D).  For instance, RNNs are often in spaces such as ℝ4 thorugh ℝ7 because of the horizontal, vertical, pixel depth, and movie frame dimensions.
Finite Math
It is unfortunate that no combination of any three books I can think of has all of these.

Directed graphs — Learn this BEFORE trees or circuits (artificial nets) because it is the superset topography of all those configurations
Abstract symbol trees (ASTs)
Advanced set theory
Decision trees
Markov chains
Chaos theory (especially the difference between random and pseudo-random)
Game Theory starting with Von Neumann and Morgenstern's Game Theory, the seminal work in that field
Convergence in discrete systems especially the application of theory to signal saturation in integer, fixed point, or floating-point arithmetic
Statistical means, deviations, correlation, and the more progressive concepts of entropy, relative entropy, and cross-entropy
Curve fitting
Convolution
Probability especially Bayes' Theorem
Algorithmic theory (Gödel's uncertainty theorems and Turing completeness)

Chemistry and Neurology
It is good to recall chemical equilibria from high school chemistry.  Balance plays a key role in more sophisticated AI designs.  Understanding the symbiotic relationship between generative and discriminative models in GANs will help a student further this understanding.
The control functions within biological systems remain a primary source of proofs of concept in artificial intelligence research.  As researchers become more creative in imagining forms of adaptation that do not directly mimic some aspect of biology (still a distance off as of this writing) creativity may play a larger role in AI research objective formulation.
Even so, AI will probably remain a largely interdisciplinary field.